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Full file at https://FratStock.euCh. 2 Trigonometric Functions

2.1 Angles and Their Measure

1 Convert between Decimals and Degrees, Minutes, Seconds Measures for Angles

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Draw the angle.1) 60°

A) B)

C) D)

2) 135°A) B)

C) D)

Full file at https://FratStock.eu3) 2π

3A) B)

C) D)

4) - 3π4A) B)

C) D)

Full file at https://FratStock.eu5) -150°

A) B)

C) D)

6) 330°A) B)

C) D)

Full file at https://FratStock.eu7) - 7π

6A) B)

C) D)

8) 5π3A) B)

C) D)

Full file at https://FratStock.eu9) -120°

A) B)

C) D)

10) 7π4A) B)

C) D)

Convert the angle to a decimal in degrees. Round the answer to two decimal places.11) 11°41ʹ45ʹʹ

A) 11.71° B) 11.76° C) 11.66° D) 11.70°

Full file at https://FratStock.eu12) 140°49ʹ59ʹʹ

A) 140.83° B) 140.79° C) 140.89° D) 140.84°

13) 270°8ʹ40ʹʹA) 270.15° B) 270.10° C) 270.20° D) 270.14°

14) 23°47ʹ37ʹʹA) 23.94° B) 23.84° C) 23.79° D) 23.52°

15) 21°17ʹ34ʹʹA) 21.22° B) 21.34° C) 21.29° D) 21.37°

Convert the angle to D° Mʹ Sʹʹ form. Round the answer to the nearest second.16) 39.08°

A) 39°4ʹ48ʹʹ B) 39°4ʹ54ʹʹ C) 39°4ʹ8ʹʹ D) 39°4ʹ36ʹʹ

17) 175.32°A) 175°20ʹ12ʹʹ B) 175°19ʹ32ʹʹ C) 175°19ʹ12ʹʹ D) 175°17ʹ32ʹʹ

18) 265.43°A) 265°26ʹ47ʹʹ B) 265°25ʹ48ʹʹ C) 265°25ʹ43ʹʹ D) 265°47ʹ43ʹʹ

2 Find the Length of an Arc of a Circle


If s denotes the length of the arc of a circle of radius r subtended by a central angle θ, find the missing quantity.1) r = 4.05 centimeters, θ = 6 radians, s = ?

A) 25.3 cm B) 23.3 cm C) 24.3 cm D) 26.3 cm

2) r = 14.0 inches, θ = 30°, s = ?A) 7.6 in. B) 7.5 in. C) 7.4 in. D) 7.3 in.

3) r = 13 feet, s = 8 feet, θ = ?

A) 83 radians B) 24° C) 8

3° D) 24 radians

4) s = 9.5 meters, θ = 2.5 radians, r = ?A) 3 m B) 3.8 m C) 0.26 m D) 1.9 m

Full file at https://FratStock.euFind the length s. Round the answer to three decimal places.

5)

s

π4

12 mA) 9.425 m B) 18.85 m C) 1.047 m D) 15.279 m

6)

π5

s

4 cmA) 3.927 cm B) 6.366 cm C) 5.026 cm D) 2.513 cm

7)

s

55°8 cm

A) 8.447 cm B) 7.679 cm C) 6.911 cm D) 6.143 cm

8)

s

30°5 m

A) 2.356 m B) 2.094 m C) 2.618 m D) 2.88 m

Solve the problem.9) For a circle of radius 4 feet, find the arc length s subtended by a central angle of 30°. Round to the nearest

hundredth.A) 6.28 ft B) 2.09 ft C) 4.19 ft D) 376.99 ft

Full file at https://FratStock.eu10) For a circle of radius 4 feet, find the arc length s subtended by a central angle of 60°. Round to the nearest

hundredth.A) 4.25 ft B) 4.35 ft C) 4.40 ft D) 4.19 ft

11) A ship in the Pacific Ocean measures its position to be 31°16ʹ north latitude. Another ship is reported to be duenorth of the first ship at 38°26ʹ north latitude. Approximately how far apart are the two ships? Round to thenearest mile. Assume that the radius of the Earth is 3960 miles.

A) 28,369 mi B) 484 mi C) 28,380 mi D) 495 mi

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

12) Salt Lake City, Utah, is due north of Flagstaff, Arizona. Find the distance between Salt Lake City (40 °45ʹ northlatitude) and Flagstaff (35°16ʹ north latitude). Assume that the radius of the Earth is 3960 miles. Round tonearest whole mile.


13) The minute hand of a clock is 6 inches long. How far does the tip of the minute hand move in 10 minutes? Ifnecessary, round the answer to two decimal places.

A) 6.28 in. B) 4.54 in. C) 8.79 in. D) 7.51 in.

14) A pendulum swings though an angle of 30° each second. If the pendulum is 45 inches long, how far does its tipmove each second? If necessary, round the answer to two decimal places.

A) 24.85 in. B) 21.71 in. C) 25.99 in. D) 23.56 in.

3 Convert from Degrees to Radians and from Radians to Degrees


Convert the angle in degrees to radians. Express the answer as multiple of π.1) 90°

A) π2

B) π8

C) π3

D) π4

2) -36°

A) - π7

B) - π5

C) - π6

D) - π4

3) 75°

A) 6π13

B) 4π11

C) 5π12

D) 12π5

4) -75°

A) - 4π11

B) - 5π12

C) - 6π13

D) - 12π5

5) 87°

A) 29π30

B) 29π90

C) 29π60

D) 29π120

6) 6°

A) π60

B) π15

C) π18

D) π30

Full file at https://FratStock.euConvert the angle in radians to degrees.

7) 6π9A) 122° B) 120° C) 119° D) 121°

8) - 5π2A) -449° B) -452° C) -450° D) -451°

9) π3A) 60π° B) 60° C) 1° D) 3°

10) - π5A) -36π° B) -36° C) -1° D) 1°

11) 7π4A) 154° B) 315° C) 630° D) 103π°

12) 83π

A) 240° B) 480° C) 8° D) 960π°

13) π6A) 15° B) 60° C) 1080° D) 30°

14) 11π12A) 150° B) 165° C) 210° D) 160°

Convert the angle in degrees to radians. Express the answer in decimal form, rounded to two decimal places.15) 45°

A) 0.78 B) 0.79 C) 0.77 D) 0.76

16) -239°A) -4.14 B) -4.15 C) -4.17 D) -4.16

Convert the angle in radians to degrees. Express the answer in decimal form, rounded to two decimal places.17) 2

A) 0.03° B) 116.07° C) 0.2° D) 114.59°

18) 8.96A) 513.22° B) 0.23° C) 0.16° D) 513.37°

19) 6A) 140.35° B) 0.04° C) -0.04° D) 141.68°

Full file at https://FratStock.eu4 Find the Area of a Sector of a Circle


If A denotes the area of the sector of a circle of radius r formed by the central angle θ, find the missing quantity. Ifnecessary, round the answer to two decimal places.

1) r = 13 inches, θ = π6 radians, A = ?

A) 44.22 in2 B) 88.44 in2 C) 6.8 in2 D) 3.4 in2

2) r = 14 feet, A = 100 square feet, θ = ?A) 1.02 radians B) 9800 radians C) 19,600 radians D) 0.51 radians

3) θ = π4 radians, A = 56 square meters, r = ?

A) 4.69 m B) 21.98 m C) 87.92 m D) 11.94 m

4) r = 7 inches, θ = 45°, A = ?A) 2.75 in2 B) 19.23 in2 C) 38.47 in2 D) 5.5 in2

5) r = 7 feet, A = 92 square feet, θ = ?A) 215.26° B) 258,420.38° C) 129,210.19° D) 107.63°

6) θ = 60°, A = 98 square meters, r = ?A) 7.16 m B) 51.29 m C) 13.68 m D) 205.15 m

7) r = 63.9 centimeters, θ = π6 radians, A = ?

A) 340.3 cm2 B) 16.7 cm2 C) 1069 cm2 D) 2138 cm2

8) r = 11.9 feet, θ = 15.361°, A = ?A) 18.98 ft2 B) 37.96 ft2 C) 21.98 ft2 D) 40.96 ft2

Find the area A. Round the answer to three decimal places.9)

π3

6 ftA) 3.142 ft2 B) 12 ft2 C) 18.85 ft2 D) 37.699 ft2

Full file at https://FratStock.eu10)

π6

12 ydA) 3.142 yd2 B) 24 yd2 C) 37.699 yd2 D) 75.398 yd2

11)

55°

10 mA) 15.278 m2 B) 47.997 m2 C) 4.8 m2 D) 95.993 m2

12)

25°10 yd

A) 43.633 yd2 B) 21.817 yd2 C) 2.182 yd2 D) 6.944 yd2

Solve the problem.13) A circle has a radius of 12 centimeters. Find the area of the sector of the circle formed by an angle of 75°. If

necessary, round the answer to two decimal places.A) 30 cm2 B) 188.5 cm2 C) 7.85 cm2 D) 94.25 cm2

14) An irrigation sprinkler in a field of lettuce sprays water over a distance of 30 feet as it rotates through an angleof 120°. What area of the field receives water? If necessary, round the answer to two decimal places.

A) 942.48 ft2 B) 1884.96 ft2 C) 300 ft2 D) 31.42 ft2

15) As part of an experiment to test different liquid fertilizers, a sprinkler has to be set to cover an area of 110square yards in the shape of a sector of a circle of radius 60 yards. Through what angle should the sprinkler beset to rotate? If necessary, round the answer to two decimal places.

A) 2.63° B) 1.75° C) 11° D) 3.5°

16) The blade of a windshield wiper sweeps out an angle of 135° in one cycle. The base of the blade is 12 inchesfrom the pivot point and the tip is 32 inches from the pivot point. What area does the wiper cover in one cycle?(Round to the nearest 0.1 square inch.)

A) 1105.3 in2 B) 1036.7 in2 C) 1041.8 in2 D) 948.3 in2

Full file at https://FratStock.eu5 Find the Linear Speed of an Object Traveling in Circular Motion


Solve the problem.

1) An object is traveling around a circle with a radius of 10 centimeters. If in 20 seconds a central angle of 13

radian is swept out, what is the linear speed of the object?

A) 16 cm/sec B) 1

6 radians/sec C) 6 radians/sec D) 6 cm/sec

2) An object is traveling around a circle with a radius of 20 meters. If in 10 seconds a central angle of 15 radian is

swept out, what is the linear speed of the object?

A) 15 m/sec B) 1

8 m/sec C) 1

4 m/sec D) 2

5 m/sec

3) An object is traveling around a circle with a radius of 10 meters. If in 15 seconds a central angle of 3 radians isswept out, what is the linear speed of the object?

A) 3 m/sec B) 23 m/sec C) 2 m/sec D) 1

3 m/sec

4) A weight hangs from a rope 20 feet long. It swings through an angle of 27° each second. How far does theweight travel each second? Round to the nearest 0.1 foot.

A) 9.4 feet B) 9.0 feet C) 8.1 feet D) 8.7 feet

5) A gear with a radius of 2 centimeters is turning at π3 radians/sec. What is the linear speed at a point on the

outer edge of the gear?

A) π6 cm/sec B) 6π cm/sec C) 2π

3 cm/sec D) 3π

2 cm/sec

6) A wheel of radius 5.2 feet is moving forward at 10 feet per second. How fast is the wheel rotating?A) 0.6 radians/sec B) 3.2 radians/sec C) 1.9 radians/sec D) 0.52 radians/sec

7) A car is traveling at 48 mph. If its tires have a diameter of 26 inches, how fast are the carʹs tires turning? Expressthe answer in revolutions per minute. If necessary, round to two decimal places.

A) 3899.08 rpm B) 1241.11 rpm C) 620.56 rpm D) 633.56 rpm

8) A pick-up truck is fitted with new tires which have a diameter of 42 inches. How fast will the pick-up truck bemoving when the wheels are rotating at 430 revolutions per minute? Express the answer in miles per hourrounded to the nearest whole number.

A) 61 mph B) 9 mph C) 27 mph D) 54 mph

9) The Earth rotates about its pole once every 24 hours. The distance from the pole to a location on Earth 53° northlatitude is about 2383.2 miles. Therefore, a location on Earth at 53° north latitude is spinning on a circle ofradius 2383.2 miles. Compute the linear speed on the surface of the Earth at 53° north latitude.

A) 99 mph B) 14,974 mph C) 601 mph D) 624 mph

10) To approximate the speed of a river, a circular paddle wheel with radius 0.68 feet is lowered into the water. Ifthe current causes the wheel to rotate at a speed of 8 revolutions per minute, what is the speed of the current? Ifnecessary, round to two decimal places.

A) 34.18 mph B) 0.39 mph C) 0.06 mph D) 0.19 mph

Full file at https://FratStock.euSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

11) The four Galilean moons of Jupiter have orbital periods and mean distances from Jupiter given by thefollowing table.

Distance (km) Period (Earth hours)Io 4.214 × 105 42.460Europa 6.709 × 105 85.243Ganymeade 1.070 × 106 171.709Callisto 1.883 × 106 400.536

Find the linear speed of each moon. Which is the fastest (in terms of linear speed)?


12) In a computer simulation, a satellite orbits around Earth at a distance from the Earthʹs surface of 2.4 x 104 miles.The orbit is circular, and one revolution around Earth takes 10.6 days. Assuming the radius of the Earth is 3960miles, find the linear speed of the satellite. Express the answer in miles per hour to the nearest whole mile.

A) 14,600 mph B) 691 mph C) 110 mph D) 593 mph

13) A carousel has a radius of 19 feet and takes 27 seconds to make one complete revolution. What is the linearspeed of the carousel at its outside edge? If necessary, round the answer to two decimal places.

A) 4.42 ft/sec B) 0.7 ft/sec C) 119.38 ft/sec D) 8.93 ft/sec

2.2 Trigonometric Functions: Unit Circle Approach

1 Find the Exact Values of the Trigonometric Functions Using a Point on the Unit Circle


In the problem, t is a real number and P = (x, y) is the point on the unit circle that corresponds to t. Find the exact valueof the indicated trigonometric function of t.

1) (38, 55

8) Find sin t.

A) 558

B) 553

C) 3 5555

D) 38

2) (49, 65

9) Find tan t.

A) 4 6565

B) 654

C) 94

D) 659

3) ( 558

, 38) Find sec t.

A) 553

B) 83

C) 3 5555

D) 8 5555

4) (- 659

, 49) Find cos t.

A) 49

B) - 659

C) - 654

D) - 9 6565

Full file at https://FratStock.eu5) (- 39

8, 58) Find cot t.

A) 58

B) - 85

C) 398

D) - 395

6) (- 116

, - 56) Find sin t.

A) - 116

B) 65

C) - 56

D) - 6 1111

7) (- 337

, - 47) Find cot t.

A) 337

B) - 334

C) 334

D) - 4 3333

8) (56, - 11

6) Find csc t.

A) - 6 1111

B) 116

C) 115

D) - 116

9) (58, - 39

8) Find cos t.

A) 398

B) 58

C) - 58

D) - 398

10) (37, - 2 10

7) Find csc t.

A) - 106

B) 73

C) 37

D) - 7 1020

2 Find the Exact Values of the Trigonometric Functions of Quadrantal Angles


Find the exact value. Do not use a calculator.1) sin 2π

A) 0 B) 22

C) 1 D) undefined

2) cos 0

A) 22

B) 0 C) 1 D) undefined

3) tan 0

A) 1 B) 0 C) 22

D) undefined

Full file at https://FratStock.eu4) cot 0

A) 0 B) 1 C) 22

D) undefined

5) cot π2

A) 1 B) 0 C) -1 D) undefined

6) tan πA) 1 B) 0 C) -1 D) undefined

7) cos πA) 0 B) -1 C) 1 D) undefined

8) cot 3π2

A) 1 B) -1 C) 0 D) undefined

9) sin (38π)A) -1 B) 0 C) 1 D) undefined

10) sin (- π2)

A) -1 B) 0 C) 1 D) undefined

11) cos (-π)A) 1 B) 0 C) -1 D) undefined

3 Find the Exact Values of the Trigonometric Functions of π/4 = 45°


Find the exact value. Do not use a calculator.

1) cos π4

A) 2 B) - 22

C) 32

D) 22

2) cos 45°

A) 32

B) 2 C) 22

D) 12

Find the exact value of the expression if θ = 45°. Do not use a calculator.3) f(θ) = sec θ Find f(θ).

A) 2 33

B) - 2 C) 2 D) 22

4) g(θ) = sin θ Find [g(θ)]2.

A) 12

B) 2 C) - 22

D) 2

Full file at https://FratStock.eu5) f(θ) = cos θ Find 3f(θ).

A) 22

B) 3 22

C) - 3 22

D) - 22

6) g(θ) = sin θ Find 6g(θ).A) -6 2 B) 6 2 C) 3 2 D) -3 2

Solve the problem.7) If friction is ignored, the time t (in seconds) required for a block to slide down an inclined plane is given by the

formula

t = 2ag sinθ cosθ

where a is the length (in feet) of the base and g ≈ 32 feet per second per second is the acceleration of gravity.How long does it take a block to slide down an inclined plane with base a = 12 when θ = 45°? If necessary,round the answer to the nearest tenth of a second.

A) 1.5 sec B) 1.3 sec C) 0.3 sec D) 1.2 sec

8) The force acting on a pendulum to bring it to its perpendicular resting point is called the restoring force. Therestoring force F, in Newtons, acting on a string pendulum is given by the formula

F = mg sinθwhere m is the mass in kilograms of the pendulumʹs bob, g ≈ 9.8 meters per second per second is theacceleration due to gravity, and θ is angle at which the pendulum is displaced from the perpendicular. What isthe value of the restoring force when m = 0.9 kilogram and θ = 45°? If necessary, round the answer to thenearest tenth of a Newton.

A) 6 N B) 6.4 N C) 7.5 N D) 6.2 N

4 Find the Exact Values of the Trigonometric Functions of π/6 = 30° and π/3 = 60°


Find the exact value. Do not use a calculator.1) cot 30°

A) 32

B) 33

C) 1 D) 3

2) csc 60°

A) 32

B) 2 C) 2 33

D) 2

3) csc π6

A) 2 B) 12

C) 2 D) 2 33

4) cot π3

A) 1 B) 33

C) 3 D) 12

Full file at https://FratStock.euFind the exact value of the expression. Do not use a calculator.

5) cot 45° - cos 30°

A) 2 3 - 3 26

B) - 36

C) 2 - 22

D) 2 - 32

6) cot 60° - cos 45°

A) 2 2 - 3 36

B) 2 - 22

C) 2 3 - 3 26

D) 2 - 32

7) cos 60° + tan 60°

A) 3 32

B) 1 + 2 32

C) 2 3 D) 1 + 32

8) sin π3 - cos π

6

A) 3 B) 1 C) 3 - 12

D) 0

9) tan π6 - cos π

6

A) - 62

B) 2 3 - 3 26

C) - 36

D) 3

Find the exact value of the expression if θ = 30°. Do not use a calculator.10) f(θ) = sin θ Find f(θ).

A) 12

B) 22

C) 32

D) 33

11) g(θ) = cos θ Find g(2θ).

A) 1 B) 3 C) 12

D) 32

12) f(θ) = sin θ Find [f(θ)]2.

A) 14

B) 34

C) 1 D) 12

13) g(θ) = sin θ Find 12g(θ).

A) 6 3 B) - 12

C) 6 D) - 32

14) f(θ) = cos θ Find 11f(θ).

A) 112

B) 11 32

C) - 12

D) - 32

Find the exact value of the expression if θ = 60°. Do not use a calculator.15) f(θ) = csc θ Find f(θ).

A) 2 B) 32

C) 2 33

D) 2

Full file at https://FratStock.eu16) g(θ) = cos θ Find [g(θ)]2.

A) 34

B) 3 C) 32

D) 14

17) f(θ) = sin θ Find 10f(θ).

A) - 12

B) 5 3 C) - 32

D) 5

18) g(θ) = cos θ Find 7g(θ).

A) 7 32

B) 72

C) - 12

D) - 32


formula

t = 2ag sinθ cosθ


A) 1.2 sec B) 1.5 sec C) 2.5 sec D) 0.4 sec



A) 4.8 N B) 2.4 N C) 4.2 N D) 2.5 N

5 Find the Exact Values for Integer Multiples of π/6 = 30°, π/4 = 45°, and π/3 = 60°


Find the exact value. Do not use a calculator.

1) cos 8π3

A) 12

B) - 32

C) 32

D) - 12

2) sec 21π4

A) -2 B) - 2 C) 22

D) - 2 33

3) sin 405°

A) - 12

B) 22

C) 12

D) - 22

Full file at https://FratStock.eu4) cot 390°

A) 33

B) 3 C) - 33

D) - 3

Find the exact value of the expression. Do not use a calculator.

5) tan 7π4 + tan 5π

4

A) 2 2 + 16

B) 12

C) 0 D) 2 + 12

6) sin 135° - sin 270°

A) 22

B) 2 + 22

C) 2 D) 2 - 22

7) cos π3 + tan 5π

3

A) 2 3 + 36

B) 3 + 33

C) 1 - 2 32

D) 3 + 12

8) cos 120° tan 60°

A) 32

B) 32

C) - 32

D) - 14

9) tan 150° cos 210°

A) 3 + 12

B) 3 3 + 2 36

C) 2 3 + 36

D) - 5 36

10) sin 330° sin 270°

A) 12

B) - 32

C) - 12

D) 32

6 Use a Calculator to Approximate the Value of a Trigonometric Function


Use a calculator to find the approximate value of the expression rounded to two decimal places.1) sin 17°

A) 0.21 B) -1.04 C) -0.96 D) 0.29

2) cos 39°A) 0.15 B) 0.78 C) 0.27 D) 0.90

3) tan 82°A) 0.33 B) 0.38 C) 7.17 D) 7.12

4) cos 3π5

A) -0.31 B) 1.00 C) 1.07 D) -0.24

Full file at https://FratStock.eu5) sec π

12A) 1.04 B) 1.13 C) 0.91 D) 1.00

6) csc 66°A) -37.66 B) 1.09 C) 1.20 D) -37.55

7) cot π8

A) 2.30 B) 2.41 C) 145.79 D) 145.90

8) cot 0.2944A) 0.30 B) 3.30 C) 0.96 D) 1.04

9) cos 2A) 1.00 B) -0.42 C) -1.00 D) 0.42

10) cos 1°A) 0.54 B) 1.00 C) -0.54 D) -1.00

11) tan 37°A) -0.84 B) 0.75 C) 0.80 D) 0.60


formula

t = 2ag sinθ cosθ


A) 1.1 sec B) 1 sec C) 0.3 sec D) 1.2 sec



A) 6.8 N B) 7 N C) 6.6 N D) 0.8 N


14) The strength S of a wooden beam with rectangular cross section is given by the formulaS = kd3 sin2 θ cos θ

where d is the diagonal length, θ the angle illustrated, and k is a constant that varies with the type of woodused.

Let d = 1 and express the strength S in terms of the constant k for θ = 45°, 50°, 55°, 60°, and 65°. Does thestrength always increase as θ gets larger?

7 Use a Circle of Radius r to Evaluate the Trigonometric Functions


A point on the terminal side of an angle θ is given. Find the exact value of the indicated trigonometric function of θ.1) (-3, -4) Find sin θ.

A) - 35

B) 35

C) 45

D) - 45

2) (-3, 4) Find cos θ.

A) 35

B) - 35

C) - 45

D) 45

3) (- 12, 13) Find cos θ.

A) 2 1313

B) - 132

C) - 3 1313

D) 133

4) (2, 3) Find tan θ.

A) 23

B) 132

C) 32

D) - 132

5) (2, 3) Find cot θ.

A) 23

B) - 132

C) 32

D) 132

6) (-1, -1) Find csc θ.A) - 2 B) -1 C) -2 D) 2

7) (-5, -1) Find sec θ.

A) 265

B) - 3 2626

C) - 26 D) - 265

Full file at https://FratStock.euSolve the problem.

8) If sin θ = 0.2, find sin (θ + π).A) -0.2 B) 0.2 C) -0.8 D) 0.8

9) If sin θ = 17, find csc θ.

A) 7 B) 67

C) - 17

D) undefined

10) A racetrack curve is banked so that the outside of the curve is slightly elevated or inclined above the inside ofthe curve. This inclination is called the elevation of the track. The maximum speed on the track in miles perhour is given byr(29000 + 41000 tan θ)

where r is the radius of the track in miles and θ is the elevation in degrees. Find the maximum speed for aracetrack with an elevation of 29° and a radius of 0.6 miles. Round to the nearest mile per hour.

A) 50,638 mph B) 176 mph C) 200 mph D) 40,067 mph

11) The path of a projectile fired at an inclination θ to the horizontal with an initial speed vo is a parabola. Therange R of the projectile, the horizontal distance that the projectile travels, is found by the formula

R =vo2 sin 2θ

g where g = 32.2 feet per second per second or g = 9.8 meters per second per second. Find the

range of a projectile fired with an initial velocity of 197 feet per second at an angle of 17° to the horizontal.Round your answer to two decimal places.

A) 704.76 ft B) 673.97 ft C) 673.87 ft D) 352.38 ft

2.3 Properties of the Trigonometric Functions

1 Determine the Domain and the Range of the Trigonometric Functions


Solve the problem.1) What is the domain of the cosine function?

A) all real numbersB) all real numbers, except integral multiples of π (180°)C) all real numbers from -1 to 1, inclusive

D) all real numbers, except odd multiples of π2 (90°)

2) For what numbers θ is f(θ) = sec θ not defined?

A) all real numbers B) integral multiples of π (180°)

C) odd multiples of π (180°) D) odd multiples of π2 (90°)

3) For what numbers θ is f(θ) = csc θ not defined?

A) all real numbers B) integral multiples of π (180°)

C) odd multiples of π2 (90°) D) odd multiples of π (180°)

Full file at https://FratStock.eu4) What is the range of the cosine function?

A) all real numbers from -1 to 1, inclusiveB) all real numbersC) all real numbers greater than or equal to 0D) all real numbers greater than or equal to 1 or less than or equal to -1

5) What is the range of the cotangent function?A) all real numbers greater than or equal to 1 or less than or equal to -1B) all real numbersC) all real numbers from -1 to 1, inclusiveD) all real numbers, except integral multiples of π(180)°

6) What is the range of the cosecant function?A) all real numbers greater than or equal to 1 or less than or equal to -1B) all real numbersC) all real numbers from -1 to 1, inclusiveD) all real numbers, except integral multiples of π(180)°

2 Determine the Period of the Trigonometric Functions


Use the fact that the trigonometric functions are periodic to find the exact value of the expression. Do not use acalculator.

1) sin 495°

A) 12

B) - 22

C) - 12

D) 22

2) tan 390°

A) 3 B) 33

C) - 3 D) 32

3) csc 660°

A) - 3 B) - 2 C) - 2 33

D) - 12

4) cot 750°

A) 3 B) - 33

C) - 3 D) 33

5) cot 720°A) 3 B) -1 C) 0 D) undefined

6) tan 720°

A) 1 B) 0 C) 33

D) undefined

7) cos 20π3

A) 12

B) - 32

C) - 12

D) 32

Full file at https://FratStock.eu8) sin 22π

3

A) - 12

B) -1 C) - 32

D) 32

9) tan 9π4

A) 3 B) 33

C) -1 D) 1

10) sec 21π4

A) 22

B) -2 C) - 2 33

D) - 2

Solve the problem.11) If cos θ = -0.8, find the value of cos θ + cos (θ + 2π) + cos (θ + 4π).

A) -0.8 B) -2.4 + 6π C) -2.4 D) -0.4

12) If tan θ = 2.9, find the value of tan θ + tan (θ + π) + tan (θ + 2π).A) 8.7 + 3π B) 10.7 C) 8.7 D) undefined

13) If f(θ) = sin θ and f(a) = 16, find the exact value of f(a) + f(a + 2π) + f(a + 4π).

A) 16

B) 12 + 6π C) 5

2D) 1

2

14) If f(θ) = cot θ and f(a) = -4, find the exact value of f(a) + f(a + π) + f(a + 3π).A) -4 B) -12 C) -12 + 4π D) undefined

15) If f(θ) = cos θ and f(a) = - 112

, find the exact value of f(a) + f(a - 2π) + f(a + 4π).

A) -12 B) -36 C) - 14

D) - 112


16) If f(θ) = sin θ and f(a) = - 19, find the exact value of f(a) + f(a - 4π) + f(a - 2π).


17) If sin θ = -0.8, find the value of sin θ + sin (θ + 2π) + sin (θ + 4π).A) -0.4 B) -2.4 + 6π C) -2.4 D) -0.8

18) If cot θ = 7.3, find the value of cot θ + cot (θ + π) + cot (θ + 2π).A) 21.9 B) 23.9 C) 21.9 + 3π D) undefined

Full file at https://FratStock.eu3 Determine the Signs of the Trigonometric Functions in a Given Quadrant


Name the quadrant in which the angle θ lies.1) tan θ > 0, sin θ < 0

A) I B) II C) III D) IV

2) cos θ < 0, csc θ < 0A) I B) II C) III D) IV

3) sin θ > 0, cos θ < 0A) I B) II C) III D) IV

4) cot θ < 0, cos θ > 0A) I B) II C) III D) IV

5) csc θ > 0, sec θ > 0A) I B) II C) III D) IV

6) sec θ < 0, tan θ < 0A) I B) II C) III D) IV

7) tan θ < 0, sin θ < 0A) I B) II C) III D) IV

8) cos θ > 0, csc θ < 0A) I B) II C) III D) IV

9) cot θ > 0, sin θ < 0A) I B) II C) III D) IV

10) sin θ > 0, cos θ > 0A) I B) II C) III D) IV

Solve the problem.11) Which of the following trigonometric values are negative?

I. sin(-292°)II. tan(-193°)III. cos(-207°)IV. cot 222°

A) II and III B) I and III C) II, III, and IV D) III only


12) Determine the sign of the trigonometric values listed below.(i) sin 250°(ii) tan 330°(iii) cos(-40°)

Full file at https://FratStock.eu4 Find the Values of the Trigonometric Functions Using Fundamental Identities


In the problem, sin θ and cos θ are given. Find the exact value of the indicated trigonometric function.

1) sin θ = 2 23

, cos θ = 13

Find tan θ.

A) 3 B) 2 2 C) 24

D) 3 24

2) sin θ = 2 23

, cos θ = 13

Find cot θ.

A) 3 24

B) 2 2 C) 3 D) 24

3) sin θ = 2 23

, cos θ = 13

Find sec θ.

A) 2 2 B) 24

C) 3 D) 3 24

4) sin θ = 53

, cos θ = 23

Find csc θ.

A) 32

B) 3 55

C) 2 55

D) 52

Use the properties of the trigonometric functions to find the exact value of the expression. Do not use a calculator.5) sin2 40° + cos2 40°

A) 1 B) 2 C) -1 D) 0

6) sec2 40° - tan2 40°A) -1 B) 1 C) 0 D) 2

7) tan 70° cot 70°A) 0 B) -1 C) 70 D) 1

8) tan 25° - sin 25°cos 25°

A) 0 B) 25 C) 1 D) undefined

5 Find Exact Values of the Trig Functions of an Angle Given One of the Functions and the Quadrant of the Angle


Find the exact value of the indicated trigonometric function of θ.

1) tan θ = - 83, θ in quadrant II Find cos θ.

A) 738

B) - 733

C) 3 7373

D) - 3 7373

Full file at https://FratStock.eu2) csc θ = - 9

8, θ in quadrant III Find cot θ.

A) - 8 1717

B) 178

C) - 179

D) - 9 1717

3) sec θ = 52 , θ in quadrant IV Find tan θ.

A) - 215

B) - 21 C) - 212

D) - 52

4) tan θ = 815

, 180°< θ < 270° Find cos θ.

A) - 1517

B) -15 2323

C) -15 D) 8 2323

5) cos θ = 725

, 3π2 < θ < 2π Find cot θ.

A) -7 26

B) - 724

C) - 247

D) 257

6) cos θ = 89, tan θ < 0 Find sin θ.

A) - 98

B) - 17 C) - 179

D) - 178

7) sin θ = - 49, tan θ > 0 Find sec θ.

A) - 4 6565

B) - 9 6565

C) 94

D) - 659

8) cot θ = - 32, cos θ < 0 Find csc θ.

A) 3 1313

B) - 133

C) 132

D) - 3 1313

9) sin θ = 12, sec θ < 0 Find cos θ and tan θ.

A) cos θ = - 3, tan θ = - 10 33

B) cos θ = - 32

, tan θ = - 33

C) cos θ = 32, tan θ = 3

3D) cos θ = - 3

2, tan θ = 3

3


10) sin θ = 16, sec θ < 0 Find cos θ and tan θ.

Full file at https://FratStock.eu6 Use Even-Odd Properties to Find the Exact Values of the Trigonometric Functions


Use the even-odd properties to find the exact value of the expression. Do not use a calculator.1) sin (-30°)

A) - 32

B) - 12

C) 32

D) 12

2) sin (-60°)

A) - 12

B) 32

C) 12

D) - 32

3) sec (-60°)

A) 2 B) 2 33

C) - 2 33

D) -2

4) cot (-60°)

A) - 33

B) 3 C) - 3 D) 33

5) cos (-150°)

A) 32

B) 12

C) -12

D) - 32

6) cos - π4

A) - 32

B) 32

C) 22

D) - 22

7) sec - π6

A) 2 33

B) 2 C) - 2 33

D) -2

8) cot - π6

A) - 33

B) 33

C) - 3 D) 3

9) sin - π2

A) 0 B) 1 C) -1 D) undefined

10) tan (-π)A) 1 B) -1 C) 0 D) undefined

Full file at https://FratStock.eu11) cot - π

4

A) 1 B) -1 C) - 3 D) - 33

Solve the problem.

12) If f(θ) = sin θ and f(a) = - 14, find the exact value of f(-a).

A) 34

B) - 14

C) 14

D) - 34

13) If f(θ) = tan θ and f(a) = 4, find the exact value of f(-a).

A) - 14

B) 14

C) -4 D) 4


14) Is the function f(θ) = sin θ + cos θ even, odd, or neither?

15) Is the function f(θ) = sin θ + tan θ even, odd, or neither?

Full file at https://FratStock.eu2.4 Graphs of the Sine and Cosine Functions

1 Graph Functions of the Form y = A sin(ωx) Using Transformations


Use transformations to graph the function.1) y = 4 sin x

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

A)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

Full file at https://FratStock.eu2) y = sin (x + π)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

A)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

Full file at https://FratStock.eu3) y = sin x + 4

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

A)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

Full file at https://FratStock.eu4) y = -5 sin x

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

A)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

Full file at https://FratStock.eu5) y = sin (πx)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

A)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

Full file at https://FratStock.eu6) y = 2 sin x + 1

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

A)

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

B)

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

C)

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

D)

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

Full file at https://FratStock.eu7) y = -3 sin (x + π

3)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

A)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

Full file at https://FratStock.eu8) y = 5 sin (π - x)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

A)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

Solve the problem.9) For what numbers x, 0 ≤ x ≤ 2π, does sin x = 0?

A) 0, 1, 2 B) 0, π, 2π C) 0, 1 D) π2, 3π2

10) For what numbers x, 0 ≤ x ≤ 2π, does sin x = 1?

A) 0, 2π B) π2, 3π2

C) π2

D) none

Full file at https://FratStock.eu11) For what numbers x, 0 ≤ x ≤ 2π, does sin x = -1?

A) 3π2

B) π C) π2, 3π2

D) none

2 Graph Functions of the Form y = A cos(ωx) Using Transformations


Use transformations to graph the function.1) y = 2 cos x

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

A)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

Full file at https://FratStock.eu2) y = cos (x - π

4)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

A)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

Full file at https://FratStock.eu3) y = cos x + 1

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

A)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

Full file at https://FratStock.eu4) y = -5 cos x

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

A)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

Full file at https://FratStock.eu5) y = cos (π

2x)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

A)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

Full file at https://FratStock.eu6) y = 3 cos x - 1

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

A)

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

B)

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

C)

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

D)

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-π π 2π 3π

y10

8

6

4

2

-2

-4

-6

-8

-10

Full file at https://FratStock.eu7) y = -4 cos (x - π

4)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

A)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

Full file at https://FratStock.eu8) y = 5 cos (π - x)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

A)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

Solve the problem.9) What is the y-intercept of y = cos x?

A) 1 B) π C) π2

D) 0

10) For what numbers x, 0 ≤ x ≤ 2π, does cos x = 0?

A) π2, 3π2

B) 0, π, 2π C) 0, 1, 2 D) 0, 1

Full file at https://FratStock.eu11) For what numbers x, 0 ≤ x ≤ 2π, does cos x = 1?

A) 0, 2π B) π2

C) π2, 3π2

D) none

12) For what numbers x, 0 ≤ x ≤ 2π, does cos x = -1?

A) π2

B) π C) π2, 3π2

D) none

3 Determine the Amplitude and Period of Sinusoidal Functions


Without graphing the function, determine its amplitude or period as requested.1) y = -3 sin x Find the amplitude.

A) -3π B) π3

C) 2π D) 3

2) y = -5 sin 12x Find the amplitude.

A) 5 B) π5

C) 4π D) 5π2

3) y = 3 sin 5x Find the amplitude.

A) π3

B) π5

C) 35

D) 3

4) y = sin 3x Find the period.

A) 2π B) 3 C) 1 D) 2π3

5) y = 3 cos 14x Find the amplitude.

A) 3 B) π3

C) 3π4

D) 8π

6) y = cos 3x Find the period.

A) 3 B) 2π C) 2π3

D) 1

7) y = 5 cos 13x Find the period.

A) π3

B) 6π C) 5π3

D) 5

8) y = -5 cos x Find the period.

A) π B) 5 C) 2π D) π5

Full file at https://FratStock.eu9) y = 9

8 cos (- 6π

5x) Find the period.

A) 9π4

B) 53

C) 49

D) 12π5

10) y = 76 cos (- 4π

5x) Find the amplitude.

A) 76

B) 6π7

C) 52

D) 4π5


Solve the problem.11) Wildlife management personnel use predator-prey equations to model the populations of certain predators

and their prey in the wild. Suppose the population M of a predator after t months is given by

M = 750 + 125 sin π6t

while the population N of its primary prey is given by

N = 12,250 + 3050 cos π6t

Find the period for each of these functions.

12) The average daily temperature T of a city in the United States is approximated by

T = 55 - 23 cos 2π365

(t -30)

where t is in days, 1 ≤ t ≤ 365, and t = 1 corresponds to January 1. Find the period of T.


13) The current I, in amperes, flowing through a particular ac (alternating current) circuit at time t seconds is I = 240 sin (70πt)

What is the period and amplitude of the current?

A) period = 135 second, amplitude = 240 B) period = 1

350 second, amplitude = 350

C) period = 70π seconds, amplitude = 135

D) period = π240

second, amplitude = 70


14) The current I, in amperes, flowing through an ac (alternating current) circuit at time t, in seconds, isI = 30 sin(50πt)

What is the amplitude? What is the period? Graph this function over two periods beginning at t = 0.

t

I

t

I

15) A mass hangs from a spring which oscillates up and down. The position P of the mass at time t is given byP = 4 cos(4t)

What is the amplitude? What is the period? Graph this function over two periods beginning at t = 0.

tπ2 π

P

4

-4

tπ2 π

P

4

-4

Full file at https://FratStock.eu16) Before exercising, an athlete measures her air flow and obtains

a = 0.65 sin 2π5

t

where a is measured in liters per second and t is the time in seconds. If a > 0, the athlete is inhaling; if a < 0, theathlete is exhaling. The time to complete one complete inhalation/exhalation sequence is a respiratory cycle. What is the amplitude? What is the period? What is the respiratory cycle? Graph a over two periods beginning at t = 0.

t5 10

a

1

-1

t5 10

a

1

-1

17) A boy is flying a model airplane while standing on a straight line. The plane, at the end of a twenty -five footwire, flies in circles around the boy. The directed distance of the plane from the straight line is found to be

d = 25 cos 3π4

t

where d is measured in feet and t is the time in seconds. If d > 0, the plane is in front of the boy; if d < 0, theplane is behind him. What is the amplitude? What is the period? Graph d over two periods beginning at t = 0.

t43

83 4 16

3

d

25

-25

t43

83 4 16

3

d

25

-25

Full file at https://FratStock.eu4 Graph Sinusoidal Functions Using Key Points


Match the given function to its graph.1) 1) y = sin x 2) y = cos x

3) y = -sin x 4) y = -cos xA B

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

C D

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

A) 1B, 2D, 3C, 4A B) 1A, 2B, 3C, 4D C) 1A, 2D, 3C, 4B D) 1C, 2A, 3B, 4D

Full file at https://FratStock.eu2) 1) y = sin 3x 2) y = 3 cos x

3) y = 3 sin x 4) y = cos 3xA B

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

C D

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

A) 1A, 2C, 3D, 4B B) 1A, 2B, 3C, 4D C) 1A, 2D, 3C, 4B D) 1B, 2D, 3C, 4A

Full file at https://FratStock.eu3) 1) y = sin (x - π

2) 2) y = cos (x + π

2)

3) y = sin (x + π2) 4) y = cos (x - π

2)

A B

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

C D

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

A) 1A, 2B, 3C, 4D B) 1B, 2D, 3C, 4A C) 1C, 2A, 3B, 4D D) 1A, 2D, 3C, 4B

Full file at https://FratStock.eu4) 1) y = 1 + sin x 2) y = 1 + cos x

3) y = -1 + sin x 4) y = -1 + cos xA B

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

C D

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

x-2π -π π 2π

y3

2

1

-1

-2

-3

A) 1A, 2C, 3D, 4B B) 1A, 2D, 3C, 4B C) 1A, 2B, 3C, 4D D) 1B, 2D, 3C, 4A

Full file at https://FratStock.eu5) 1) y = sin ( 1

3x) 2) y = 1

3 cos x

3) y = 13 sin x 4) y = cos ( 1

3x)

A B

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

C D

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

A) 1A, 2D, 3C, 4B B) 1A, 2B, 3C, 4D C) 1A, 2C, 3D, 4B D) 1B, 2D, 3C, 4A

Full file at https://FratStock.eu6) 1) y = -2 sin (2x) 2) y = -2 sin ( 1

2x)

3) y = 2 cos (2x) 4) y = 2 cos ( 12x)

A B

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

C D

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

A) 1C, 2A, 3B, 4D B) 1C, 2A, 3D, 4B C) 1D, 2B, 3A, 4C D) 1A, 2C, 3D, 4B

Full file at https://FratStock.eu7) 1) y = -3 sin (π

3x) 2) y = -3 sin ( 1

3x)

3) y = -3 cos (π3x) 4) y = -3 cos ( 1

3x)

A) B)

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

C) D)

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

A) 1A, 2C, 3B, 4D B) 1C, 2A, 3D, 4B C) 1B, 2D, 3A, 4C D) 1A, 2C, 3D, 4B

Graph the sinusoidal function.8) y = 3 sin (πx)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

Full file at https://FratStock.euA)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

9) y = -3 cos (πx)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6


x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

10) y = 3 sin (2x)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6


x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

11) y = 3 cos (πx)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6


x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

12) y = -4 sin ( 14x)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6


x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

B)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

C)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

D)

x-π π 2π 3π

y6

4

2

-2

-4

-6

x-π π 2π 3π

y6

4

2

-2

-4

-6

13) y = 74 cos (- 1

2x)

x-2π 2π 4π 6π 8π

y6

4

2

-2

-4

-6

x-2π 2π 4π 6π 8π

y6

4

2

-2

-4

-6


x-2π 2π 4π 6π 8π

y6

4

2

-2

-4

-6

x-2π 2π 4π 6π 8π

y6

4

2

-2

-4

-6

B)

x-2π 2π 4π 6π 8π

y6

4

2

-2

-4

-6

x-2π 2π 4π 6π 8π

y6

4

2

-2

-4

-6

C)

x-2π 2π 4π 6π 8π

y6

4

2

-2

-4

-6

x-2π 2π 4π 6π 8π

y6

4

2

-2

-4

-6

D)

x-2π 2π 4π 6π 8π

y6

4

2

-2

-4

-6

x-2π 2π 4π 6π 8π

y6

4

2

-2

-4

-6

Answer the question.14) Which one of the equations below matches the graph?

A) y = 3 cos 14x B) y = 3 sin 1

4x C) y = 3 cos 4x D) y = -3 sin 4x

Full file at https://FratStock.eu15) Which one of the equations below matches the graph?

A) y = 4 sin 12x B) y = 4 cos 1

2x C) y = 4 cos 2x D) y = 2 cos 1

4x

16) Which one of the equations below matches the graph?

A) y = 2 sin 13x B) y = 2 cos 1

3x C) y = 2 cos 3x D) y = -2 sin 1

3x

17) Which one of the equations below matches the graph?

A) y = -2 sin 3x B) y = 2 sin 13x C) y = -2 cos 3x D) y = -2 sin 1

3x

5 Find an Equation for a Sinusoidal Graph


Write the equation of a sine function that has the given characteristics.1) Amplitude: 5

Period: 6π

A) y = 5 sin (6x) B) y = sin (6x) + 5 C) y = 5 sin 13x D) y = 6 sin 2

5x

Full file at https://FratStock.eu2) Amplitude: 5

Period: 3

A) y = 5 sin 23πx B) y = 3 sin 2

5πx C) y = 5 sin (3x) D) y = sin (3πx) + 5

Find an equation for the graph.3)

x-2π -π π 2π

y54321

-1-2-3-4-5

x-2π -π π 2π

y54321

-1-2-3-4-5

A) y = 2 sin (3x) B) y = 2 sin 13x C) y = 3 sin (2x) D) y = 3 sin 1

2x

4)

x-2π -π π 2π

y54321

-1-2-3-4-5

x-2π -π π 2π

y54321

-1-2-3-4-5

A) y = 3 cos (2x) B) y = 3 cos 12x C) y = 2 cos 1

3x D) y = 2 cos (3x)

5)

x-2π -π π 2π

y54321

-1-2-3-4-5

x-2π -π π 2π

y54321

-1-2-3-4-5

A) y = -4 sin (2x) B) y = -4 cos (2x) C) y = -4 sin 12x D) y = -4 cos 1

2x


x-2π -π π 2π

y54321

-1-2-3-4-5

x-2π -π π 2π

y54321

-1-2-3-4-5

A) y = 2 sin 13x B) y = 2 sin (3x) C) y = 3 sin 1

2x D) y = 3 sin (2x)

7)

x-2π -π π 2π

y54321

-1-2-3-4-5

x-2π -π π 2π

y54321

-1-2-3-4-5

A) y = 4 cos (2x) B) y = 2 cos (4x) C) y = 4 cos 12x D) y = 2 cos 1

4x

8)

x-2π -π π 2π

y54321

-1-2-3-4-5

x-2π -π π 2π

y54321

-1-2-3-4-5

A) y = -4 sin (2x) B) y = -4 cos (2x) C) y = -4 cos 12x D) y = -4 sin 1

2x


x-12

12

y54321

-1-2-3-4-5

x-12

12

y54321

-1-2-3-4-5

A) y = 5 sin π2x B) y = 2 sin π

5x C) y = 2 sin (5πx) D) y = 5 sin (2πx)

10)

x-5 -4 -3 -2 -1 1 2 3 4 5 6

y54321

-1-2-3-4-5

x-5 -4 -3 -2 -1 1 2 3 4 5 6

y54321

-1-2-3-4-5

A) y = 4 cos (2πx) B) y = 2 cos π4x C) y = 4 cos π

2x D) y = 2 cos (4πx)

11)

A) y = -4 cos (2x) B) y = 4 cos 12x C) y = 4 sin (2x) D) y = 4 cos (2x)


A) y = -5 sin (3x) B) y = 5 cos 13x C) y = -5 sin 2

3x D) y = -5 sin 1

3x

13)

A) y = -3 sin (3x) B) y = -3 cos (3x) C) y = -3 cos 13x D) y = 3 cos 1

3x

14)

A) y = 12 cos (4x) B) y = 1

2 cos 1

2x C) y = cos (4x) D) y = 1

2 cos 1

4x

Full file at https://FratStock.eu2.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions

1 Graph Functions of the Form y = A tan(ωx) + B and y = A cot(ωx) + B


Match the function to its graph.1) y = tan x

A)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

B)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

C)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

D)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

Full file at https://FratStock.eu2) y = tan x + π

2A)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

B)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

C)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3

D)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3

Full file at https://FratStock.eu3) y = tan (x + π)

A)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

B)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

C)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

D)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

Full file at https://FratStock.eu4) y = tan x - π

2A)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

B)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

C)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3

D)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3

Graph the function.5) y = -cot x

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3


x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

B)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

C)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3

D)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3

6) y = tan (x - π)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3


x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

B)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

x-π -π

2π2 π

3π2 2π 5π

2 3π

y3

-3

C)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3

D)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3

x-π -π

2π2 π 3π

2 2π 5π2 3π

y3

-3

7) y = 3 tan (4x)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6


x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

B)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

C)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

D)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

8) y = 2 tan 14x

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6


x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

B)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

C)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

D)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

9) y = -cot (πx)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6


x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

B)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

C)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

D)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

10) y = cot(2x)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6


x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

B)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

C)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

D)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

11) y = -2 cot (4x)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6


x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

B)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

C)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

D)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

12) y = -3 tan x + π4

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6


x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

B)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

C)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

D)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

13) y = 12 cot x + π

4

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6


x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

B)

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π

3π2 2π 5π

2 3π

y6

4

2

-2

-4

-6

C)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

D)

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

x-π -π

2π2 π 3π

2 2π 5π2 3π

y6

4

2

-2

-4

-6

14) y = -tan x - π4

x-7 7

y

7

-7

x-7 7

y

7

-7


x-7 7

y

7

-7

x-7 7

y

7

-7

B)

x-7 7

y

7

-7

x-7 7

y

7

-7

C)

x-7 7

y7

-7

x-7 7

y7

-7

D)

x-7 7

y7

-7

x-7 7

y7

-7

15) y = tan x - π4

x-7 7

y7

-7

x-7 7

y7

-7


x-7 7

y7

-7

x-7 7

y7

-7

B)

x-7 7

y7

-7

x-7 7

y7

-7

C)

x-7 7

y7

-7

x-7 7

y7

-7

D)

x-7 7

y7

-7

x-7 7

y7

-7

16) y = -cot x - π4

x-7 7

y7

-7

x-7 7

y7

-7


x-7 7

y7

-7

x-7 7

y7

-7

B)

x-7 7

y7

-7

x-7 7

y7

-7

C)

x-7 7

y7

-7

x-7 7

y7

-7

D)

x-7 7

y7

-7

x-7 7

y7

-7

Solve the problem.17) What is the y-intercept of y = sec x?

A) π2

B) 1 C) 0 D) none

18) What is the y-intercept of y = cot x?

A) 0 B) 1 C) π2

D) none

19) For what numbers x, -2π ≤ x ≤ 2π, does the graph of y = tan x have vertical asymptotes.?

A) -2π, -π, 0, π, 2π B) - 3π2

, - π2, π2, 3π2

C) -2, -1, 0, 1, 2 D) none

20) For what numbers x, -2π ≤ x ≤ 2π, does the graph of y = csc x have vertical asymptotes?

A) -2, -1, 0, 1, 2 B) - 3π2

, - π2, π2, 3π2

C) -2π, -π, 0, π, 2π D) none

Full file at https://FratStock.eu2 Graph Functions of the Form y = A csc(ωx) + B and y = A sec(ωx) + B


Graph the function.

1) y = csc x + π3

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

A)

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

B)

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

C)

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

D)

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

Full file at https://FratStock.eu2) y = sec x + π

6

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

A)

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

B)

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

C)

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

D)

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

Full file at https://FratStock.eu3) y = -sec x

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

A)

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

B)

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

C)

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

D)

x-2π -π π 2π

y3

-3

x-2π -π π 2π

y3

-3

Full file at https://FratStock.eu4) y = csc (3x)

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

A)

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

B)

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

C)

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

D)

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

Full file at https://FratStock.eu5) y = csc 1

4x

x-2π 2π 4π 6π 8π

y3

-3

x-2π 2π 4π 6π 8π

y3

-3

A)

x-2π 2π 4π 6π 8π

y3

-3

x-2π 2π 4π 6π 8π

y3

-3

B)

x-2π 2π 4π 6π 8π

y3

-3

x-2π 2π 4π 6π 8π

y3

-3

C)

x-2π 2π 4π 6π 8π

y3

-3

x-2π 2π 4π 6π 8π

y3

-3

D)

x-2π 2π 4π 6π 8π

y3

-3

x-2π 2π 4π 6π 8π

y3

-3

Full file at https://FratStock.eu6) y = 2 csc 1

3x

x-2π 2π 4π 6π 8π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π 2π 4π 6π 8π

y10

8

6

4

2

-2

-4

-6

-8

-10

A)

x-2π 2π 4π 6π 8π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π 2π 4π 6π 8π

y10

8

6

4

2

-2

-4

-6

-8

-10

B)

x-2π 2π 4π 6π 8π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π 2π 4π 6π 8π

y10

8

6

4

2

-2

-4

-6

-8

-10

C)

x-2π 2π 4π 6π 8π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π 2π 4π 6π 8π

y10

8

6

4

2

-2

-4

-6

-8

-10

D)

x-2π 2π 4π 6π 8π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π 2π 4π 6π 8π

y10

8

6

4

2

-2

-4

-6

-8

-10

Full file at https://FratStock.eu7) y = 2 sec (3x)

x-π -π

2π2 π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-π -π

2π2 π

y10

8

6

4

2

-2

-4

-6

-8

-10

A)

x-π -π

2π2 π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-π -π

2π2 π

y10

8

6

4

2

-2

-4

-6

-8

-10

B)

x-π -π

2π2 π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-π -π

2π2 π

y10

8

6

4

2

-2

-4

-6

-8

-10

C)

x-π -π

2π2 π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-π -π

2π2 π

y10

8

6

4

2

-2

-4

-6

-8

-10

D)

x-π -π

2π2 π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-π -π

2π2 π

y10

8

6

4

2

-2

-4

-6

-8

-10

Full file at https://FratStock.eu8) y = 6 csc x + π

2

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

A)

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

B)

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

C)

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

D)

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

Full file at https://FratStock.eu9) y = -4 sec x - π

4

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

A)

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

B)

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

C)

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

D)

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

x-2π -π π 2π

y10

8

6

4

2

-2

-4

-6

-8

-10

Full file at https://FratStock.eu10) y = csc πx

5 + 3π

5

x-10 10

y10

-10

x-10 10

y10

-10

A)

x-10 10

y10

-10

x-10 10

y10

-10

B)

x-10 10

y10

-10

x-10 10

y10

-10

C)

x-10 10

y10

-10

x-10 10

y10

-10

D)

x-10 10

y10

-10

x-10 10

y10

-10

Solve the problem.11) A rotating beacon is located 5 ft from a wall. If the distance from the beacon to the point on the wall where the

beacon is aimed is given bya = 5|sec 2πt|,

where t is in seconds, find a when t = 0.29 seconds. Round your answer to the nearest hundredth.A) 14.13 ft B) 8.16 ft C) -20.11 ft D) 20.11 ft

Full file at https://FratStock.eu2.6 Phase Shift; Sinusoidal Curve Fitting

1 Graph Sinusoidal Functions of the Form y = A sin (ωx -φ) + B


Find the phase shift of the function.

1) y = -3 sin x - π2

A) π2 units to the right B) -3 units up

C) π2 units to the left D) -3 units down

2) y = -5 cos x + π2

A) π2 units to the right B) -5 units up

C) -5 units down D) π2 units to the left

3) y = -4 sin 4x - π2

A) π8 units to the right B) 4π units up

C) 4π units down D) π2 units to the left

4) y = 2 cos (6x + π)

A) 2π units to the right B) 6π units to the right

C) π2 units to the left D) π

6 units to the left

5) y = -3 sin 14x - π

4

A) π16 units to the left B) π

4 units to the right

C) π3 units to the left D) π units to the right

6) y = 5 cos 12x + π

2

A) π2 units to the left B) π


C) π units to the left D) 5π units to the right

Full file at https://FratStock.eu7) y = 4 sin (4πx - 3)

A) 3 units to the left B) 3 units to the right

C) 34 units to the left D) 3

4π units to the right

8) y = 4 sin -3x - π2

A) π6 units to the right B) π


C) π2 units to the left D) π

6 units to the left

Graph the function. Show at least one period.9) y = 2 sin (4πx + 2)

x-1 1

y4

3

2

1

-1

-2

-3

-4

x-1 1

y4

3

2

1

-1

-2

-3

-4

A)

x-1 1

y4

3

2

1

-1

-2

-3

-4

x-1 1

y4

3

2

1

-1

-2

-3

-4

B)

x-1 1

y4

3

2

1

-1

-2

-3

-4

x-1 1

y4

3

2

1

-1

-2

-3

-4

Full file at https://FratStock.euC)

x-1 1

y4

3

2

1

-1

-2

-3

-4

x-1 1

y4

3

2

1

-1

-2

-3

-4

D)

x-1 1

y4

3

2

1

-1

-2

-3

-4

x-1 1

y4

3

2

1

-1

-2

-3

-4

10) y = 5 sin(3x - π)

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

A)

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

B)

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8


x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

D)

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

11) y = 2 cos 4x + π2

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

A)

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

B)

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8


x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

D)

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

12) y = -3 sin 4x + π2

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

A)

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

B)

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8


x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

D)

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

13) y = 3 sin(πx + 5)

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

A)

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

B)

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8


x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

D)

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

14) y = 4 cos -5x + π2

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

A)

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

B)

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8


x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

D)

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

x-π π 2π 3π

y8

6

4

2

-2

-4

-6

-8

15) y = 4 sin(-2x - π)

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

A)

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

B)

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5


x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

D)

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

16) y = -3 cos(x - π)

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

A)

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

B)

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5


x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

D)

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

x-2π 2π

y5

4

3

2

1

-1

-2

-3

-4

-5

Solve the problem.

17) For the equation y = - 12 sin(4x + 3π), identify (i) the amplitude, (ii) the phase shift, and (iii) the period.

A) (i) - 12

(ii) - 4π3 (iii) 4 B) (i) 1

2(ii) - 3π

4(iii) 4

C) (i) 2 (ii) 3π (iii) π2

D) (i) 12

(ii) - 3π4

(iii) π2

18) For the equation y = - 12 cos(2x - 2π), identify (i) the amplitude, (ii) the phase shift, and (iii) the period.

A) (i) 2 (ii) π (iii) π B) (i) 12

(ii) π2

(iii) π

C) (i) 2 (ii) 2π (iii) 2π D) (i) 12

(ii) π (iii) π

Write the equation of a sine function that has the given characteristics.19) Amplitude: 3

Period: 4π

Phase Shift: π4

A) y = 3 sin 2x + 18π B) y = 3 sin 1

2x + 1

8π

C) y = 3 sin 4x + π4

D) y = 3 sin 12x - 1

8π

20) Amplitude: 3Period: 5π

Phase Shift: - π5

A) y = 3 sin 52x - 2

25π B) y = 3 sin 2

5x - 2

25π

C) y = 3 sin 5x - π5

D) y = 3 sin 25x + 2

25π

Full file at https://FratStock.eu21) Amplitude: 3

Period: πPhase Shift: - 2

A) y = 3 sin 12x - 4 B) y = 3 sin (2x + 4) C) y = sin (3x + 2) D) y = 3 sin (x - 2)

22) Amplitude: 5Period: π

Phase Shift: 72

A) y = sin (5x + 7) B) y = 5 sin 12x - 14 C) y = 5 sin (2x - 7) D) y = 5 sin 2x + 7

2

2 Build Sinusoidal Models from Data


Solve the problem.1) An experiment in a wind tunnel generates cyclic waves. The following data is collected for 56 seconds:

Time (in seconds)

Wind speed(in feet per second)

0 1514 4328 7142 4356 15

Let V represent the wind speed (velocity) in feet per second and let t represent the time in seconds. Write a sineequation that describes the wave.

A) V = 71 sin π28

t - π2 + 15 B) V = 71 sin(56t - 28) + 15

C) V = 28 sin π28

t - π2 + 43 D) V = 56 sin (56t - 28) + 28

Full file at https://FratStock.eu2) A townʹs average monthly temperature data is represented in the table below:

Month, x Average Monthly Temperature, °F

January, 1February, 2March, 3April, 4May, 5June, 6July, 7August, 8September, 9October, 10November, 11 December, 12

25.128.941.956.571.181.086.181.083.056.638.928.3

Find a sinusoidal function of the form y = A sin (ωx - φ) + B that fits the data.

A) y = 55.6 sin π6x - π

4 + 30.5 B) y = 25.1 sin π

6x - π

4 + 86.1

C) y = 30.5 sin π6x - 2π

3 + 55.6 D) y = 86.1 sin π

6x - 2π

3 + 25.1

3) The number of hours of sunlight in a day can be modeled by a sinusoidal function. In the northern hemisphere,the longest day of the year occurs at the summer solstice and the shortest day occurs at the winter solstice. In2000, these dates were June 22 (the 172nd day of the year) and December 21 (the 356th day of the year),respectively.

A town experiences 10.95 hours of sunlight at the summer solstice and 8.48 hours of sunlight at the wintersolstice. Find a sinusoidal function y = A sin (ωx - φ) + B that fits the data, where x is the day of the year. (Note:There are 366 days in the year 2000.)

A) y = 1.235 sin π183

x - 2π3 + 9.715 B) y = 1.235 sin π

183x - 161π

366 + 9.715

C) y = 10.95 sin 172π356

x - 2π3 + 9.715 D) y = 10.95 sin πx - 2π

3 + 8.48

Full file at https://FratStock.eu4) The data below represent the average monthly cost of natural gas in an Oregon home.

Month Aug Sep Oct Nov Dec JanCost 21.20 28.24 44.73 67.25 89.77 106.26

Month Feb Mar Apr May Jun JulCost 111.30 106.26 89.77 67.25 43.73 28.24

Above is the graph of 45.05 sin x superimposed over a scatter diagram of the data. Find the sinusoidal functionof the form y = A sin (ωx - φ) + B which best fits the data.

A) y = 45.05 sin π4x - 2π

3 + 21.20 B) y = 45.05 sin π

8t + 12 + 21.20

C) y = 45.05 sin π6x - π

12 + 66.25 D) y = 45.05 sin π

6x - 2π

3 + 66.25


5) The data below represent the average monthly cost of natural gas in an Oregon home.

Month Aug Sep Oct Nov Dec JanCost 18.90 24.24 44.58 68.25 91.92 109.26

Month Feb Mar Apr May Jun JulCost 113.60 106.26 91.92 68.25 42.58 24.24

Above is the graph of 47.35 sin x. Make a scatter diagram of the data. Find the sinusoidal function of the formy = A sin (ωx - φ) + B which fits the data.

Full file at https://FratStock.eu6) The following data represents the normal monthly precipitation for a certain city in California.

Month, xNormal Monthly

Precipitation, inchesJanuary, 1February, 2March, 3April, 4May, 5June, 6July, 7August, 8September, 9October, 10November, 11December, 12

6.064.454.382.081.270.560.170.460.912.245.215.51

Draw a scatter diagram of the data for one period. Find a sinusoidal function of the form y = A sin (ωx - φ) + Bthat fits the data. Draw the sinusoidal function on the scatter diagram. Use a graphing utility to find thesinusoidal function of best fit. Draw the sinusoidal function of best fit on the scatter diagram.

x

y

x

y

Full file at https://FratStock.eu7) The following data represents the normal monthly precipitation for a certain city in Arkansas.

Month, xNormal Monthly

Precipitation, inchesJanuary, 1February, 2March, 3April, 4May, 5June, 6July, 7August, 8September, 9October, 10November, 11December, 12

3.914.365.316.217.027.848.198.067.416.305.214.28

Draw a scatter diagram of the data for one period. Find the sinusoidal function of the formy = A sin (ωx - φ) + B that fits the data. Draw the sinusoidal function on the scatter diagram. Use a graphingutility to find the sinusoidal function of best fit. Draw the sinusoidal function of best fit on the scatter diagram.

x

y

x

y

Full file at https://FratStock.eu8) The following data represents the average monthly minimum temperature for a certain city in California.

Month, x

Average Monthly Minimum

Temperature, °FJanuary, 1February, 2March, 3April, 4May, 5June, 6July, 7August, 8September, 9October, 10November, 11December, 12

49.650.855.657.560.763.665.965.664.462.154.250.1

Draw a scatter diagram of the data for one period. Find a sinusoidal function of the form y = A sin (ωx - φ) + Bthat fits the data. Draw the sinusoidal function on the scatter diagram. Use a graphing utility to find thesinusoidal function of best fit. Draw the sinusoidal function of best fit on the scatter diagram.

x

y

x

y

Full file at https://FratStock.eu9) The following data represents the average percent of possible sunshine for a certain city in Indiana.

Month, x Average Percent of Possible Sunshine

January, 1February, 2March, 3April, 4May, 5June, 6July, 7August, 8September, 9October, 10November, 11December, 12

465155606873757468624138

Draw a scatter diagram of the data for one period. Find the sinusoidal function of the formy = A sin (ωx - φ) + B that fits the data. Draw the sinusoidal function on the scatter diagram. Use a graphingutility to find the sinusoidal function of best fit. Draw the sinusoidal function of best fit on the scatter diagram.

x

y

x

y

Full file at https://FratStock.euCh. 2 Trigonometric FunctionsAnswer Key

2.1 Angles and Their Measure1 Convert between Decimals and Degrees, Minutes, Seconds Measures for Angles

1) B2) C3) A4) D5) B6) C7) D8) D9) C10) D11) D12) A13) D14) C15) C16) A17) C18) B

2 Find the Length of an Arc of a Circle1) C2) D3) D4) B5) A6) D7) B8) C9) B10) D11) D12) 379 mi13) A14) D

3 Convert from Degrees to Radians and from Radians to Degrees1) A2) B3) C4) B5) C6) D7) B8) C9) B10) B11) B12) B13) D14) B15) B

Full file at https://FratStock.eu16) C17) D18) D19) A

4 Find the Area of a Sector of a Circle1) A2) A3) D4) B5) A6) C7) C8) A9) C10) C11) B12) B13) D14) A15) D16) B

5 Find the Linear Speed of an Object Traveling in Circular Motion1) A2) D3) C4) A5) C6) C7) C8) D9) D10) B11) 6.24 × 104 kmp; 4.95 × 104 kmp; 3.92 × 104 kmp; 2.95 × 104 kmp; Io12) B13) A

2.2 Trigonometric Functions: Unit Circle Approach1 Find the Exact Values of the Trigonometric Functions Using a Point on the Unit Circle

1) A2) B3) D4) B5) D6) C7) C8) A9) B10) D

2 Find the Exact Values of the Trigonometric Functions of Quadrantal Angles1) A2) C3) B4) D5) B6) B

Full file at https://FratStock.eu7) B8) C9) B10) A11) C

3 Find the Exact Values of the Trigonometric Functions of π/4 = 45°1) D2) C3) C4) A5) B6) C7) D8) D

4 Find the Exact Values of the Trigonometric Functions of π/6 = 30° and π/3 = 60°1) D2) C3) C4) B5) D6) C7) B8) D9) C10) A11) C12) A13) C14) B15) C16) D17) B18) B19) B20) D

5 Find the Exact Values for Integer Multiples of π/6 = 30°, π/4 = 45°, and π/3 = 60°1) D2) B3) B4) B5) C6) B7) C8) C9) D10) A

6 Use a Calculator to Approximate the Value of a Trigonometric Function1) D2) B3) D4) A5) A6) B7) B

Full file at https://FratStock.eu8) B9) B10) B11) B12) A13) A14) 0.354k; 0.377k; 0.385k; 0.375k and 0.347k; No, it reaches a maximum near 55 °.

7 Use a Circle of Radius r to Evaluate the Trigonometric Functions1) D2) B3) C4) C5) A6) A7) D8) A9) A10) B11) B

2.3 Properties of the Trigonometric Functions1 Determine the Domain and the Range of the Trigonometric Functions

1) A2) D3) B4) A5) B6) A

2 Determine the Period of the Trigonometric Functions1) D2) B3) C4) A5) D6) B7) C8) C9) D10) D11) C12) C13) D14) B15) C

16) - 13

17) C18) A

3 Determine the Signs of the Trigonometric Functions in a Given Quadrant1) C2) C3) B4) D5) A6) B

Full file at https://FratStock.eu7) D8) D9) C10) A11) A12) (i) negative

(ii) negative(iii) positive

4 Find the Values of the Trigonometric Functions Using Fundamental Identities1) B2) D3) C4) B5) A6) B7) D8) A

5 Find Exact Values of the Trig Functions of an Angle Given One of the Functions and the Quadrant of the Angle1) D2) B3) C4) A5) B6) C7) B8) C9) B

10) cos θ = - 356

, tan θ = - 3535

6 Use Even-Odd Properties to Find the Exact Values of the Trigonometric Functions1) B2) D3) A4) A5) D6) C7) A8) C9) C10) C11) B12) C13) C14) neither15) odd

2.4 Graphs of the Sine and Cosine Functions1 Graph Functions of the Form y = A sin(ωx) Using Transformations

1) B2) A3) D4) D5) D6) D7) A

Full file at https://FratStock.eu8) B9) B10) C11) A

2 Graph Functions of the Form y = A cos(ωx) Using Transformations1) D2) B3) D4) B5) B6) B7) D8) A9) A10) A11) A12) B

3 Determine the Amplitude and Period of Sinusoidal Functions1) D2) A3) D4) D5) A6) C7) B8) C9) B10) A11) 12, 1212) 365 days13) A

14) amplitude = 30, period = 125

I = 30sin(50πt)

t125

225

I

30

-30

t125

225

I

30

-30

Full file at https://FratStock.eu15) amplitude = 4, period = π

2

P = 4cos(4t)

tπ2 π

P

4

-4

tπ2 π

P

4

-4

16) amplitude = 0.65, period = 5, respiratory cycle = 5 seconds

a = 0.65sin 2π5

t

t5 10

a

0.65

-0.65

t5 10

a

0.65

-0.65

17) amplitude = 25, period = 8/3

d = 25 cos 3π4

t

t83

163

d

25

-25

t83

163

d

25

-25

Full file at https://FratStock.eu4 Graph Sinusoidal Functions Using Key Points

1) D2) D3) C4) D5) D6) D7) C8) D9) A10) A11) D12) C13) A14) A15) C16) A17) A

5 Find an Equation for a Sinusoidal Graph1) C2) A3) D4) C5) C6) B7) A8) B9) D10) C11) D12) D13) C14) A

2.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions1 Graph Functions of the Form y = A tan(ωx) + B and y = A cot(ωx) + B

1) A2) D3) C4) A5) D6) D7) D8) A9) B10) D11) B12) B13) D14) D15) C16) C17) B18) D19) B

Full file at https://FratStock.eu20) C

2 Graph Functions of the Form y = A csc(ωx) + B and y = A sec(ωx) + B1) B2) A3) D4) C5) D6) A7) A8) A9) B10) C11) D

2.6 Phase Shift; Sinusoidal Curve Fitting1 Graph Sinusoidal Functions of the Form y = A sin (ωx -φ) + B

1) A2) D3) A4) D5) D6) C7) D8) D9) C10) A11) B12) C13) D14) A15) B16) C17) D18) D19) D20) D21) B22) C

2 Build Sinusoidal Models from Data1) C2) C3) B4) D


y = 47.35 sin ( π6x - 2π

3) + 66.25

6) y = 3.14 sin (0.46x + 1.52) + 3.16

7) y = 2.17 sin (0.49x - 1.88) + 6.02

8) y = 8.33 sin (0.50x - 2.06) + 57.97

9) y = 15.99 sin (0.57x - 2.29) + 60.62

full file at ://fratstock.eu/sample/test-bank-trigonometry-a...full file at ch. 2 trigonometric...

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